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Geometry multi-step problem

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Geometry multi-step problem

by ctgambler » Tue Jun 23, 2009 6:50 pm
Here's a question that got me stuck...if you can answer it, please explain each step.

The U.S. Defense Department has decided that the Pentagon is an obsolete building and that it must be replaced with an upgraded version: the Hexagon. The Secretary of Defense wants a building that is exactly 70 feet high and 200 feet on a side, and taht has a hexagonal bulls-eye cutout in the center that is 50 feet on a side. What will be the volume of the new building in cubic feet?
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by yogami » Tue Jun 23, 2009 7:38 pm
Well if you visualize this 3D object it will look like a hexagon with another hexagon inside the hexagon. The inner hexagon is just empty and the remaining is solid. If you join all the vertices of the hexagon to the center of the hexagon you will notice that you are actually creating six trapezoid prisms on the solid object. In other words the solid object can be visualized as six regular trapezoid prisms next to each other. The volume of a trapezoid prism is the surface area times the height of it. Now the surface area or the trapezoid is (a + b)h/2 where h is the height of the surface area (not the trapezoid itself) and a and b are the lengths of the two sides of the trapezoid. We know that the longer length is 200 and the shorter length is 50. To find the height of the trapezoid surface area you first find the length of the hypotenuse of the equilateral triangles formed by joining the vertices of the outer hexagon to its center. Then you find the hypotenuse of the equilateral triangle formed by joining the vertices of the inner hexagon to the center. The difference in lengths of these hypotenuses is the height of the surface area of the trapezoid. Thus with these ridiculous lengthy calculations you finally get the surface area of the trapezoid. Now the volume of the trapezoid prism is the product of this surface area and height of the prism which is given as 70 feet. Once you find this volume you multiply it by 6 because there are six such prisms. Then you can move this question to the 780+ section because ts unlikely in GMAT.
I could have done the calculations but I have to apologize for not doing so since this is probably not going to be in GMAT
200 or 800. It don't matter no more.
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